THE POWER SPECTRA OF THE EXCITATION AND THE RESPONSE

Suppose that the response у and the excitation / are connected by the linear differential equation

(4)

Iff(t) represents a stationary fluctuation, it does not tend to zero as t oo and its Fourier transform does not exist. To overcome this difficulty the truncated function used in the last section can be used here again. Let f{t) be truncated in such a manner that it is zero outside an interval (— T, T). The Fourier transform of the truncated Д/) can be defined provided that /(/) is absolutely integrable in (— T, T) and has bounded variations. Let

 

zf(co) = – L [T fit) e-^dt (5)

V2tt J-t

When A(o) is given by Eq. 5, the functions f(t) and y(t) given by Eqs. 2 and 3 will represent, respectively, the excitation and the response within the interval (— T, T). From Eq. 3, we can derive the value of yt) in

 

the same interval. Comparing Eq. 3 with Eq. 5 of § 8.5, and following the reasoning of that section, we obtain

THE POWER SPECTRA OF THE EXCITATION AND THE RESPONSE(6)

where p(co) is the power spectrum of fit). Equation 6 shows that the power spectrum of the response is equal to that of the forcing function divided by the square of the absolute value of the impedance. The phase relationship between the excitation and the response, represented by the argument of the complex number Z(/co), is completely obliterated in the power-spectra relations.

Equation 6 may be applied to general dynamic systems for which the impedance Z(ico) can be defined. It holds generally for all dynamic systems described by differential equations, linear integral equations and linear integral-differential equations that occur in aeroelasticity.

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